Chisquare {base}

R Documentation

The (non-central) Chi-Squared Distribution

Description

Density, distribution function, quantile function and random generation for the chi-squared (\chi^2) distribution with df degrees of freedom and optional non-centrality parameter ncp.

Usage

dchisq(x, df, ncp=0, log = FALSE)
pchisq(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
qchisq(p, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
rchisq(n, df, ncp=0)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
df	degrees of freedom.
ncp	non-centrality parameter. For rnchisq, ncp=0 is the only possible value.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].

Details

The chi-squared distribution with df= n degrees of freedom has density

f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}

for x > 0. The mean and variance are n and 2n.

The non-central chi-squared distribution with df= n degrees of freedom and non-centrality parameter ncp = \lambda has density

f(x) = e^{-\lambda / 2} \sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)

for x \ge 0. It is the distribution of the sum of squares of n normals each with variance one, \lambda being the sum of squares of the normal means.

Value

dchisq gives the density, pchisq gives the distribution function, qchisq gives the quantile function, and rchisq generates random deviates.

See Also

dgamma for the Gamma distribution which generalizes the chi-squared one.

Examples

dchisq(1, df=1:3)
pchisq(1, df= 3)
pchisq(1, df= 3, ncp = 0:4)# includes the above

x <- 1:10
## Chi-squared(df = 2) is a special exponential distribution
all.equal(dchisq(x, df=2), dexp(x, 1/2))
all.equal(pchisq(x, df=2), pexp(x, 1/2))

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